If you know any other problem, please put in the comment section.

  1. Let (a_k) be a sequence in (0, 1) . Prove that a_k to 0 iff \displaystyle{ \frac{1}{n} \sum _{k=1}^n to 0 }
  2. Let f: setR to setR be a continuously differentiable function such that \infty_{x \in setR} f'(x) > 0 . Prove that there exist some a \in set R such that f(a) = 0.
  3. Let X be a metric space such that A \subset X , A is an uncountable subset of X . Prove that X is not separable. Given {d(x, y) : x \in A , y \in A } > 0
  4. Let the differential equation be y” + P(x) y’ + Q(x) y = R(x)  where P, Q, R are continuous functions on [a, b] .